Comparing Fractions that contain epsilon

Question

Given $\epsilon$ a constant s.t. $0<\epsilon<1$, and $n,p$ positive integers, $n >= 2p$, is the following true:

$\frac{(1+\epsilon)n}{(2+\epsilon)p} \geq \lceil\frac{n}{2p}\rceil$

Answer 1

No, the ceiling can make it fail. Take $n=3,p=1$ the right side is $2$, the left about $3/2$